An extremal basis method in minimax problems

220

Y. F. Dem >anov

For comparison we give the conditions determining the sets of reachability for this system in the absence of noise, that is, for v(t) =O, t=O, 1,. . . , T-l: x’(1)

21,

x2(i)

s’(Z)

31,

s”(2) 21, X+($0)>1,

5’ (10) >1,

21,

s’(1)

+x2(1) G4,

r’(2) +x2(2) GS, z’(10) +22(10) G2048,

T=l,

T=2, T=lO.

On the information hypothesis 2) the set st={a(T)ls’(T)H,

zz(T)>7}

for this system for T = 4 is reachable from the set of initial states defined by the conditions 4x’(o)+422(0) 27,

,l5x’(0) 416.~~(0) >22,

16x1 (0) +15x2 (0) 220.

In particular, it is reachable from the point x(O) = (1, 1). In conclusion the author thanks Yu. P. Ivanilov for useful advice and comments. Translated by J. Berry

REFERENCES 1.

LOTOV, A. V., A numerical method for constructing

2.

POSPELOV, I. G., Numerical methods for solving linear multi-step games. ZLr.vychisl. Mat. mut Fiz., 15,3,615-626,1975.

3,

GALE, D., Theory of linear economic models (Teoriya lineinykh ekonomicheskikh

sets of attainability for linear controlled systems with phase constraints. Zh. vychisl. Mat. mat. Fiz., 15, 1,61-78, 1915.

modelei), Izd-vo,

in. lit., Moscow, 1963. 4.

CHERNHCOV, S. N., Linear inequalities (tmeinye

5,

LOTOV, A. V., The exact discovery of the general non-negative solution of a system of homogeneous linear inequalities. In: Algorithms and programs (Algoritmy i programmy), No. 2,11-l 2, VNTITSentr, Moscow, 1972.

neravenstva), “Nauka”, Moscow, 1968.

AN EXTREMAL BASIS METHOD IN MINIMAX PROBLEMS* V. F. DEM’YANOV Leningrad {Received 1 August 1975;Revised

I December 1976)

THE PROBLEM of the minimization (on a finite-dimensional set) of a function of a maximum, taken with respect to a large (or continuous) set of functions, is reduced to a sequence of problems of the minimization of auxiliary functions, each of which is a maximum with respect to a small number of functions. 1. The function cp(W= maxf(X, Y), YE@ *Zk @hid. Mat. mat. Fiz., 17,2,512-517,

1977.

Short communications

221

is given, whereC_cE,is a compact set, the functionf(X, Y) is convex with respect to X, continuous with respect to X and Y, twice differentiable with respect to X, the vector function af/axand the matrix function @f/8X2 are continuous with respect to X and Y on E,XG, where an m>Oand a Kcmexist such that

It is required to find min v(X). TEE ?I It is known [ 1, 21 that for the point X’=E,to be an exact minimum of the function cp(X) it is necessary and sufficient that

(2)

OEL(x’),

where H(X)=

L(X)=coH(X),

{

af(f;y)1Ydw}

,

R(W=O’=Glf(X, Y)=cpWl. With every point xod,

there is associated a set L (X0). If OeL(X,), then the direction

g(Xo)= -

2(X0) -9

IlZ (X0) II

where 112(X0) II= min Il.% ZEUXO)

is the direction of steepest descent of the function I&X) at the point Xo. The ability to calculate the direction of steepest descent or directions close to it makes it possible to develop numerical methods for minimizing q(X) (see [l] ). However, it is here necessary to take into account all the points of the set R(X) and to minimize the function cp(X) on a ray (in the descent direction). In practice the set G (if it contains an infinite number of points) is discretized (replaced by a finite set of points), but a more accurate approximation of q(X) requires a large number of points of the set G to be taken into account. It would be desirable to develop a method of minimizing the function p(X), in whose realization a comparatively small number of points in the set G is ignored. A similar situation holds in the theory of Chebyshev approximation, where in [3] a method was proposed which uses at each step only n + 2 points (here n is the degree of an algebraic polynomial). The idea of the method of [3] goes back to [4]. In [3] the fact of the presence of Chebyshev altemance is essentially used. We first describe the “fundamental algorithm” (in the terminology of [5] ). We choose an arbitrary point &=E,, and an arbitrary subset of n + 2 points of the set G: uo={Yol. . . . , Yo. n+2),

Yo,EG Vi=[l:

n-t2]

We call u. the basis subset. Suppose we have already constructed the point X*sE,, and the basis ‘%={Ykir . . . , yk, n+2},

YkiEG

Vi=

[ 1 : TL+~].

V. F. Dam yanov

222 We determine the function @k(X)=

max

f(x,

(3)

Ykd

i‘S[!:n+z] and find the point Xk+l such that (PkfXk+d=

(4)

pk(x)-

mh XCiE

n

The solution of problem (4) is simpler than the minimization of the function 9(X), since to find the function 9(X) it is necessary to calculate the values of the functionf(X, Y) at n t 2 points. By the necessary condition for a minimum of the function 9&(X) the point X, satisfies the condition

(5)

OELkr where

yki)

Rk={ijf(Xk+ir

=(pk(Xk+l)}-

Since any point of the set Lk can ae represented as a convex combination of not more than n t 1 points of the set Hk (see [l] , p. 306), there exists at least one index ikE [ 1: n+2], which “can be done ~thout” (that is, either ik@Rk, or the origin of coordinates in (5) can be formed Ykik)/8x.bt the point f?kEG be such that without the VeCtOr$f(Xk+l, f

that is,

Ykd%(&+l).

we

now

(xk+h

yk)=(P(xk+I)

COnStIUCt

Y k+l,i

=

new basis

a

=

max YE0

(6)

f (xk+i, Y)?

@k+i={Yk+I,

yki,

ieik,

yk;o

t=fk.

,,

. . . ,

Yk+&

*+*}

as

fOllOWS:

‘i’hebasis ok+1 differs from the basis ok by one point (and also contains R + 2 points). If =rpkfXk+i),then xk+l is a point of mourn of the function 9(X) (since in this case f(Xk+l,Yki)=(P(Xk+l),fIZaf is, Y?+ieR(Xk+i),andby (5) the IleWSSary and sufficient condition (2) is satisfied). But if cp(Xk+l))(~k(Xk+t),then starting from the point xk+l and the basis ok+lwe construct the function (pk+f (X) and continue as before.

v(Xk+d

As a result we have a sequence of points (Xk). If this sequence is finite, then the last point obtained is by construction a point of minimum of the function 9(X). But if the sequence @h] is infinite then the following theorem holds. Theorem 1 The point sequence

{xk}

converges to a point of minimum of the function 9(X).

Roof: We first note that the sequence

{xk}

is bounded. This follows from the fact that

223

Short communications

from the uniform boundedness with respect to k of all the sets

and from the inclusion &+i=&. Ihe uniform boundedness of the set Dk follows from (1) and the inequality j(x,+A, Y)>f(Xo,Y)-KIIAII+mllAl12~fo-~iiAII+~~~A~~2, where f0 = min f (X0, Y) . YEG

We put D= ;

DR.

h-i

As already mentioned, the set D is bounded. We write &=&+l-Xk. f txk+i,

yk--i,i)

1 +

For id&_,

We

-

2 (

=f(Xk+Ak,

a'f(Ek, Ah,

yh-id)

By (5) there exists a YA--i,iE~k

yk4.i)

+

(Xk,

Yk-1.i)

3X

(

,

Ati

(8)

Ak

ax2

have f(&, Yk-,, i) =(ph-1

=f (&,

yk-1,i)

We have af

1. (xk).

(preCiSdy

Ok,

af(Xkr

yk4.i)

and

not

such that

ok__1)

(9)

ax From (1) and (8) for such an i we have 1

f (-%t+l?YA_1.f) )(pA--l (XA) +

7

m&c’s

that is,

qA(xktl)=

maX ,

ft (XA+it

YAd)

(PA--t(XA)

+

ymAA'+

(10)

This implies that &-‘o.

A-r-

(11)

Indeed, let us assume the contrary: let a number DO and a subsequence {k,} exist such that II&J AZ. Then from (10) we have (Pk(Xk+ L) -+-,butcp(Xk+i)~(PA(Xkt~),therefore (PC%)-++“, which is impossible since {Xh) is a bounded sequence and v(X) is a continuous function. Therefore (11) is proved. From (6) we have

(12)

224

jwnov

V. F. Dem

Since f

therefore

i’?dXk+d

Y)

(xk+i,

-cP(Xh)

af(gkt y)

=f(&+Ak, Y)=f(X,, Y) +

Ivk(Xk+l)

1 GKitAkll,

I(Pk(Xk+l)--dXk)

-(pk(Xk)

>

,

and from (12) we obtain

1 G:Kitbkil,

IGI(Pk(Xk+i)-(pk(Xk)

,Ak

ax

I+l~k(xk)--cp(xk)

1G=lbkii.

(13)

From this and (5) we conclude that the point Xk+l is an ek-stationary point of the function q(X), where 8k=2Klbk/. (see [l] and section 2 of this paper). Since Ek+o, then (see [l]), the sequence of ek-stationary points converges to a point of minimum of the function q(X). The theorem is proved. 2. The auxiliary problem (4) in the method explained above is not in general solved after a finite number of steps; therefore in practice the method of section 1 is unrealizable. Below we explain three more methods, two of which in fact enable us to realize the method of successive approximation explained above. We first describe the e-method. We choose a sequence

{Ek}

such that

OI

ek+ +o,

ekema

z

(14)

k-0

We take an arbitrary point X0&?, and an arbitrary (n + 2)point basis ~o={Yo1,. . . , Yo, n+2),

Y,iEG

Vi= [1 : n+2].

Let us already have constructed the point &C?%,and the basis

~k’=={Yklr

. . . , yk,

YkiEG

n+2},

Vi~[l

1 n+2].

We put

(Pktx)= max f (x, Ykf) iE’[i:n+t]

(15)

and find a point X,, 1 such that

(16)

@=Lkea,

where he,<

=

&ck=(iE

co

H kek =

Hkek,

[i

: n+2]

If(Xk+I,

ykf)

af(xk+I,

aAx Yki)=(Pk(Xk+i)}.

The point Xk+ I is called an ek-stationary point of the function cpk(X). We now construct a new basis CQ+~just as in section 1. If (F(X~+~) =(P~(X~+~) and (5) holds, then the point Xk+l is a point of minimum of the function q(X), and the process terminates. Otherwise, starting from the point Xk+r and the basis ok+1 we construct the function .(~k+l(X)’ and find an ek+ 1-stationary point of this function. If the sequence of points {x,) thus constructed is finite, then the last point obtained is a point of minimum of the function q(X). Otherwise the following theorem holds.

225

Short communications

77reorem 2 The sequence of points {Xkjconverges to a point of m~imum of the function g(X). hoot As in the proof of Theorem 1 we establish the boundedness of the sequence (xh). We again put Ah=Xk+-&.By (16) we find a Yn_,, i = ok? such that (9) is satisfied. Since for LE&--~,Er;-lwehave f(&, Yk-,, j)~:fpk-_i(Xk)-&k_-ttfrOm this and (8) we obtain f(Xk+i,

Yk-1,

i)~~k-l(Xk)-&k-~+m~,,2/2,

that is, (17) We show that A,+-+0.

(18)

Let us assume the contrary. Then we can find a number Q>O and a subsequence of subscripts {k,}such that lIArSi! aa. From (17) and (14) we conclude that (pk(Xx.&-++m, and therefore also that q (&) + + a:, which is impossible. Therefore (18) is proved. From (6) we again have ( 12) and (13). Therefore Xk+l is an yk-stationary point of the function p(X), where ~k=2Kbk\l+Ek-l. But ek-+O,therefore the sequence {Xk} also converges to a point of minimum of the function \o(X). ~ern~k I. On changing the basis in the choice of Yk a point can be found for which instead of (6) the inequality (19) is satisfied, where uh--+ +O. In the proof of Theorem 2 relations (12) and (13) are then correspondingly changed. It is simpler to find a point satisfying (19) than to find a point satisfying (6). 3. Unfortunately the e-method explained in section 2 is still not completely satisfactory, since in a number of cases finding e-stationary points requires an infmite procedure. The p-method explained below is free from this drawback. Let the sequence

(pk)

be such that

Let the point X, and the basis @k have already been found and the function (pk(X) have been determined (see (15)). We find a point X,+ 1 such that (21) Here Lk is a set defined after relation (5). The new basis ok+ f is constructed as in section 1. As a result we have a sequence CG}. Theorem 3 The sequence of points (Xhjconverges to a point of ~nimum of the function p(X).

V. F. Dem jtanov

226

The proof is similar to the proof of Theorem 2. In place of the inequality (17) in this case from (8) and (21) we obtain q)k(Xk+i) a(pk-1 (xk) - P,+_i+ y

1

mAh2.

From (20) we now easily obtain (18). We proceed as in the proof of Theorem 2. Remark 2. The choice of the point Yk on changing the basis can be made as indicated in

Remark 1. 4. Finally, we explain the E, p-method which is a combination of the p-method and the e-method. Let the sequences (ek) and (p$ satisfy relations (14) and (20). As Xk+l we choose a point for which

(22) where LKsliis a set defined in accordance with the inclusion (16). The change of basis is carried out as above. The point Xk+l satisfying the inequality (22) is also found in a finite number of steps. The convergence theorem also holds for the sequence obtained in this way. Remark 3. In the proof of Theorems I-3 the use of the strict convexity of the function f(X, Y) with respect to X (inequa~ty (1)) is essential. Without the condition a ‘jamming” effect is possible (for linear functions it is easy to construct a corresponding example). Remark 4. In the R-algorithm (see [I] , chap. I) for the solution of the problem of Chebyshev polynomial approximation the replacement of the whole basis is performed ~m~taneously. In the methods explained above it is also tempting to change the whole basis at once. Remark 5. If the basis is replaced by the rule described in Remark 1, but we then take pN

=uk>o, then in all th e methods explained the sequence of points {Xk} will converge to a kstationary point of the function p(X). Remark 6. The methods explained above generalize to cases where the minimization is

performed on a set defined by inequalities. The idea of the reduction of a problem with an infinite number of constraints to a sequence of problems with a finite number of constraints was stated and used in f63 (for linear problems) and [7,8] (for non-linear minimax problems). The number of constraints in the auxiliary problems then increased with each step. Remark 7. Since every convex function can be represented as a function of the maximum of its supporting functions (that is, every convex function is a function of the maximum of linear functions), and every convex set is representable as the intersection of half-spaces, then the extremal basis method described above can be used to soIve the problem of the m~i~zation of a convex function on a convex set, it being necessary at each step to solve a linear progra~g problem of small dimensions. Special methods can be developed to remove the possible cycling caused by the fact that the linear functions are not strictly convex.

Therefore, unlike Kelley’s method [6], the number of constraints in the auxiliary linear progr~g problem is not increased. Other methods of counteract~g the increase in dimensions are based either on Wolf’s idea [9], whose application requires special conditions to be satisfied

Short communications

227

at each step, or on the idea of discarding “inessential” constraints (see, for example, [lo] ), which in general may be accompanied by an increase in the number of constraints. Translated by J. Berry REFERENCES 1.

DEM’YANOV, V. F. and MALOZEMOV, V. N., Introduction “Nauka”, Moscow, 1972.

to the minimax (Vvedenie v minimaks),

2.

PSHENICHNYI, B. N., The necessary conditions for an extremum (Neobkhodimye “Nauka”, Moscow, 1969.

3.

REMEZ, E. Ya., Principles of numerical methods of Chebyshev approximation chebyshevskogo priblizheniya), “Naukova dumka”, Kiev, 1969.

4.

VALEE POUSSIN, J. C., Sur la methode d’approximation

usloviya ekstremuma),

(Osnovy chislennykh metodov

minimum. Ann. Sot. scient. Bruxehes,

1910-

1911,1-16. 5.

POLAK, E., Numerical methods of optimization (a unified approach) (Chislennye metody optimizatsii (edinyi podkhod)), “Mu”, Moscow, 1974.

6.

KELLEY, J. E., The cutting plane method for solving convex programs. SIAMJ.

Appl, Math., 8.4,

703-712,196O.

7.

AKILOV, G. P. and RUBINOV, A. M., The method of successive approximations for finding the polynomial of best approximation. Dokl. Akad. Nauk SSSR, 157,3,503-505, 1964.

8.

SALMON, D. M., Minimax controller design. 9-th Joint Automat. Control Conf Ann. Arbor, Michigan, 1968. Preprintsof technical papers, 495-500, New York, N. Y., 1968.

9.

POWELL, M. J. D., The minimax solution of linear equations subject to bound on the variables. Proc IVManitoba Conf Numer. Math., 1974,53-107, Winnipeg, Manitoba, 1975.

10. GEOFFRION, A. M., Relaxation and the dual method in mathematical programming. Working paper 135, Western Management Sci. Inst. Los Angeles, Univ. California, 1968.

APPLICATION OF THE BRANCH AND BOUND METHOD TO SOME DISCRETE PROGRAMMING PROBLEMS* I. G. MITEV Sofia, Bulgaria (Received 1 August 1975)

AN ALGORITHM of branch and bound type is presented for a partially integer-valued linear programming problem and for a discrete programming problem in which the variables are replaced by groups.

1. A compact branch and bound algorithm

Many computer realizations of branch and bound type algorithms are known. ln carrying out experimental calculations the question of the economy of the operative memory is not an immediate one, but in the solution of actual applied problems it becomes very important. The realization of a branch and bound algorithm for a problem of partly integer-valued linear programming in two versions is described in [ 1,2] . In the first version the computational *Zh. @chisl

Mot. mat. Fiz., 17, 2.518-522,

1977.